For more info on Euler Sequences, notation and convention see the generic entry on Euler angle sequences.
R 123 ( ϕ , θ , ψ ) = R 3 ( ψ ) R 2 ( θ ) R 1 ( ϕ ) {\displaystyle R_{123}(\phi ,\theta ,\psi )=R_{3}(\psi )R_{2}(\theta )R_{1}(\phi )} \\
The rotation matrices are
R 3 ( ψ ) = [ c ψ s ψ 0 − s ψ c ψ 0 0 0 1 ] {\displaystyle R_{3}(\psi )=\left[{\begin{matrix}c_{\psi }&s_{\psi }&0\\-s_{\psi }&c_{\psi }&0\\0&0&1\end{matrix}}\right]}
R 2 ( θ ) = [ c θ 0 − s θ 0 1 0 s θ 0 c θ ] {\displaystyle R_{2}(\theta )=\left[{\begin{matrix}c_{\theta }&0&-s_{\theta }\\0&1&0\\s_{\theta }&0&c_{\theta }\end{matrix}}\right]}
R 1 ( ϕ ) = [ 1 0 0 0 c ϕ s ϕ 0 − s ϕ c ϕ ] {\displaystyle R_{1}(\phi )=\left[{\begin{matrix}1&0&0\\0&c_{\phi }&s_{\phi }\\0&-s_{\phi }&c_{\phi }\end{matrix}}\right]}
Carrying out the matrix multiplication from right to left
R 2 ( θ ) R 1 ( ϕ ) = [ c θ 0 − s θ 0 1 0 s θ 0 c θ ] [ 1 0 0 0 c ϕ s ϕ 0 − s ϕ c ϕ ] = [ c θ s θ s ϕ − s θ c ϕ 0 c ϕ s ϕ s θ − c θ s ϕ c θ c ϕ ] {\displaystyle R_{2}(\theta )R_{1}(\phi )=\left[{\begin{matrix}c_{\theta }&0&-s_{\theta }\\0&1&0\\s_{\theta }&0&c_{\theta }\end{matrix}}\right]\left[{\begin{matrix}1&0&0\\0&c_{\phi }&s_{\phi }\\0&-s_{\phi }&c_{\phi }\end{matrix}}\right]=\left[{\begin{matrix}c_{\theta }&s_{\theta }s_{\phi }&-s_{\theta }c_{\phi }\\0&c_{\phi }&s_{\phi }\\s_{\theta }&-c_{\theta }s_{\phi }&c_{\theta }c_{\phi }\end{matrix}}\right]}
Finaly leaving us with the Euler 123 sequence
R 3 ( ψ ) R 2 ( θ ) R 1 ( ϕ ) = [ c ψ c θ c ψ s θ s ϕ + s ψ c ϕ − c ψ s θ c ϕ + s ψ s ϕ − s ψ c θ − s ψ s θ s ϕ + c ψ c ϕ s ψ s θ c ϕ + c ψ s ϕ s θ − c θ s ϕ c θ c ϕ ] {\displaystyle R_{3}(\psi )R_{2}(\theta )R_{1}(\phi )=\left[{\begin{matrix}c_{\psi }c_{\theta }&c_{\psi }s_{\theta }s_{\phi }+s_{\psi }c_{\phi }&-c_{\psi }s_{\theta }c_{\phi }+s_{\psi }s_{\phi }\\-s_{\psi }c_{\theta }&-s_{\psi }s_{\theta }s_{\phi }+c_{\psi }c_{\phi }&s_{\psi }s_{\theta }c_{\phi }+c_{\psi }s_{\phi }\\s_{\theta }&-c_{\theta }s_{\phi }&c_{\theta }c_{\phi }\end{matrix}}\right]}
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![{\displaystyle R_{3}(\psi )=\left[{\begin{matrix}c_{\psi }&s_{\psi }&0\\-s_{\psi }&c_{\psi }&0\\0&0&1\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63e956480bacaa2f9661e0c5d7211e895c3f3dd4)
![{\displaystyle R_{2}(\theta )=\left[{\begin{matrix}c_{\theta }&0&-s_{\theta }\\0&1&0\\s_{\theta }&0&c_{\theta }\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd2d14d893a246b3de1a21325b8d03813e80664e)
![{\displaystyle R_{1}(\phi )=\left[{\begin{matrix}1&0&0\\0&c_{\phi }&s_{\phi }\\0&-s_{\phi }&c_{\phi }\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/192ac3559eaf3aee6d189ef8923fa25721b238a0)
![{\displaystyle R_{2}(\theta )R_{1}(\phi )=\left[{\begin{matrix}c_{\theta }&0&-s_{\theta }\\0&1&0\\s_{\theta }&0&c_{\theta }\end{matrix}}\right]\left[{\begin{matrix}1&0&0\\0&c_{\phi }&s_{\phi }\\0&-s_{\phi }&c_{\phi }\end{matrix}}\right]=\left[{\begin{matrix}c_{\theta }&s_{\theta }s_{\phi }&-s_{\theta }c_{\phi }\\0&c_{\phi }&s_{\phi }\\s_{\theta }&-c_{\theta }s_{\phi }&c_{\theta }c_{\phi }\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdd34645c0119afceb4171b92248188c067be766)
![{\displaystyle R_{3}(\psi )R_{2}(\theta )R_{1}(\phi )=\left[{\begin{matrix}c_{\psi }c_{\theta }&c_{\psi }s_{\theta }s_{\phi }+s_{\psi }c_{\phi }&-c_{\psi }s_{\theta }c_{\phi }+s_{\psi }s_{\phi }\\-s_{\psi }c_{\theta }&-s_{\psi }s_{\theta }s_{\phi }+c_{\psi }c_{\phi }&s_{\psi }s_{\theta }c_{\phi }+c_{\psi }s_{\phi }\\s_{\theta }&-c_{\theta }s_{\phi }&c_{\theta }c_{\phi }\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/900b6e9d145c6dc81e2bdb6ed8cbad81f680b2ec)
